• A simple closed curve made up of only line segments is called a polygon. • A diagonal of a polygon is a line segment connecting two non- consecutive vertices. • A convex polygon is a polygon in which no portion of its any diagonal is in its exterior. • A quadrilateral is a polygon having only four sides. •A regular polygon is a polygon whose all sides are equal and also all angles are equal. • The sum of interior angles of a polygon of n sides is (n-2) straight angles. • The sum of interior angles of a quadrilateral is 360°. • The sum of exterior angles, taken in an order, of a polygon is 360°. • Trapezium is a quadrilateral in which a pair of opposite sides is parallel. • Kite is a quadrilateral which has two pairs of equal consecutive sides. • A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. 12/04/18 • A rhombus is a parallelogram in which adjacent sides are equal. • A rectangle is a parallelogram in which one angle is of 900. • A square is a parallelogram in which adjacent sides are equal and one angle is of 900. • In a parallelogram, opposite sides are equal, opposite angles are equal and diagonals bisect each other. • In a rhombus diagonals intersect at right angles. • In a rectangle diagonals are equal. • Five measurements can determine a quadrilateral uniquely. • A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal are given. • A quadrilateral can be constructed uniquely if the lengths of its three sides and two diagonals are given. • A quadrilateral can be constructed uniquely if its two adjacent sides and three angles are given. • A quadrilateral can be constructed uniquely if its three sides and two included angles are given. In examples 1 to 8, there are four options out of which one is correct. Write the correct answer. Example 1 : The number of diagonals in a polygon of n sides is nn( 1) nn( 2) nn( 3) (a) (b) (c) (d) n (n–3). 2 2 2 Solution : The correct answer is (c). Example 2 : The angles of a quadrilateral ABCD taken in an order are in the ratio 3 : 7 : 6 : 4. Then ABCD is a (a) kite (b) parallelogram (c) rhombus (d) trapezium Solution : The correct answer is (d). 12/04/18 Example 3 : If the diagonals of a quadrilateral bisect each other at right angles, it will be a (a) rhombus (b) trapezium (c) rectangle (d) kite Solution : The correct answer is (a). Example 4 : The sum of the angles of a quadrilateral is (a) 180° (b) 270° (c) 360° (d) 300° Solution : The correct answer is (c). Example 5 : In a square ABCD, the diagonals meet at point O. The ∆AOB is (a) isosceles right triangle (b) equilateral triangle (c) isosceles triangle but not right triangle (d) scalene right triangle. Solution : The correct answer is (a). Quadrilaterals with certain properties are given additional names. A trapezium has exactly 1 pair of parallel sides. A parallelogram has 2 pairs of parallel sides. A rectangle has 4 right angles. A rhombus has 4 congruent sides. A square has 4 congruent sides and 4 right angles. 12/04/18 Example 6 : ABCD is a quadrilateral in which AB = 5 cm, CD = 8 cm and the sum of angle A and angle D is 180°. What is the name of this quadrilateral? (a) Parallelogram (b) Trapezium (c) Rhombus (d) Can not be determined Solution : The correct answer is (b). Example 7 : Rukmini has a farm land which is triangular in shape. What is the sum of all the exterior angles taken in an order of the farm land? (a) 90° (b) 180° (c) 360° (d) Can not be determined. Solution : The correct answer is (c). Example 8 : How many sides does an octagon have? (A) 7 (b) 8 (c) 9 (d) 10 Solution : The correct answer is (b) In examples 9 and 13, fill in the blanks to make the statements true. Example 9 : The diagonals of a rhombus bisect each other at _____ angles. Solution : Right. Example 10 : For getting diagonals through vertex A of a pentagon ABCDE, A is joined to _________. Solution : C and D. Example 11 : For constructing a unique quadrilateral at least __________ measurements are required. Solution : Five. Example 12 : If diagonals of a quadrilateral bisect at right angles it is a __________. Solution : Rhombus (or square). Example 13 : The diagonals of a __________ intersect at right angles. Solution : Kite. 12/04/18 In examples 14 to 23, state whether the statements are true (T) or false (F). Example 14 : Every rectangle is a parallelogram. Solution : True. Example 15 : Every rhombus is a kite. Solution : True. Example 16 : Every parallelogram is a trapezuim. Solution : True. Example 17 : Every kite is a trapezium. Solution : False. Example 18 : Every kite is a parallelogram. Solution : False. Example 19 : Diagonals of a rectangle are perpendicular to each other. Solution : False. Example 20 : For constructing a unique parallelogram lengths of only two sides should be given. Solution : False. Diagonals of a — Parallelogram bisect each other ? bisect each other ? Rhombus are perpendicular to each other ? are equal Rectangle ? bisect each other ? bisect each other are perpendicular to Square each other are equal 12/04/18 Example 21 : is a simple closed curve. Solution : False. Example 22 : is a concave polygon. Solution : True. Example 23 : A triangle is not a polygon. Solution : False. Example 24 : The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. Is ∠ADQ + ∠CBP = ∠A + ∠C? Give reason. D Solution : Join AC, then Q C ∠CBP = ∠BCA + ∠BAC and ∠ADQ = ∠ACD + ∠DAC (Exterior angles of triangles) A B P Therefore, ∠CBP + ∠ADQ = ∠BCA + ∠BAC + ∠ACD + ∠DAC = (∠BCA + ∠ACD) + (∠BAC + ∠DAC) = ∠C + ∠A Angles in a Quadrilateral A diagonal of a quadrilateral is a segment that joins two vertices of the quadrilateral but is not a side. You can use a diagonal of a quadrilateral to show that the sum of the angle measures in a quadrilateral is 360°. Cut a quadrilateral The sum of the angle Quadrilateral with along a diagonal to measures in each 2 pairs of parallel form two triangles. triangle is 180°. sides. 12/04/18 Example 25 : If AM and CN are perpendiculars on the diagonal BD of a parallelogram ABCD, Is ∆AMD ≅ ∆CNB? Give reason. Solution : In triangles AMD and CNB, AD = BC (opposite sides of parallelogram) ∠AMB = ∠CNB = 900 ∠ADM = ∠NBC (AD || BC and BD is transversal.) So, ∆AMD ≅ ∆CNB (AAS) Example 26 : Construct a quadrilateral ABCD in which AB = AD = 5cm, BC = CD = 7cm and BD = 6cm. What type of quadrilateral is this? Solution : Looking at the rough figure, draw a line segment BD = 6cm. Taking B and D as centres and 5 cm radius, draw arcs to intersect at the point A, then taking B and D as centres and 7 cm radius, draw arcs in the opposite side of A to intersect at the point C. Join AB, AD and BC, DC. Then ABCD is the required quadrilateral. It is a kite. 12/04/18 Example 27 : Find x in the following figure. Solution : In the given figure ∠1 + 90° = 180° (linear pair) ∠1 = 90° Now, sum of exterior angles of a polygon is 360°, therefore, x + 60° + 90° + 90° + 40° = 360° x + 280° = 360° x = 80° Classifying Plane Figures Triangle Trapezoid Parallelogram Closed figure with 3 Quadrilateral with Quadrilateral with straight sides that 1 pair of parallel 2 pairs of parallel connect 3 points sides sides Rhombus Rectangle Square Circle Set of all points Parallelogram Parallelogram Parallelogram in a plane that with 4 right with 4 sides of with 4 sides of are at the same equal length angles equal length and 4 right angles distance from a fixed point 12/04/18 Example 28 : Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures. Solution : Let the angles be 4x and 5x. Then, 4x + 5x = 180° 9x = 180° x = 20° So, angles are 4 × 20° = 80° and 5 × 20° =100°. Example 29 : The four angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the angles. Solution : Let angles be 3x, 4x, 5x, 6x. Thus, 3x + 4x + 5x + 6x = 360° since sum of the angles of a quadrilateral is 360°. So, 18x = 360° or, x = 20° Thus, angles are 60°, 80°, 100°, 120°. Example 30 : In a parallelogram PQRS, the bisectors of ∠P and ∠Q meet at O. Find ∠POQ. Solution : Since OP and OQ are the bisectors of ∠P and ∠Q respectively (see figure on the right), 1 1 so, ∠OPQ = ∠P and ∠OQP = ∠Q 2 2 In ∆POQ, ∠OPQ + ∠PQO + ∠POQ = 180° (Angle sum property) 1 1 i.e. ∠P + ∠POQ + ∠Q = 180° 2 2 1 i.e. ∠POQ = 180° – (∠P + ∠Q) 2 1 = 180° – × 180° 2 = 90° 12/04/18 Example 31 : Three angles of a quadrilateral are 50°, 40° and 123°. Find its fourth angle. Solution : Let fourth angle be x. Then 500 + 400 + 1230 + x = 3600. or x = 3600 – 500 – 400 – 1230 = 3600 – 2130 = 1470. A quadrilateral is a closed plane figure with four sides that are line segments. The figures below are special types of quadrilaterals. Special Quadrilaterals Diagram Trapezium A trapezium is a quadrilateral with exactly 1 pair of parallel sides. Parallelogram A Parallelogram is a quadrilateral with 2 pairs of parallel sides. Rhombus A rhombus is a parallelogram with 4 sides of equal length. Rectangle A rectangle is a parallelogram with 4 right angles.